For most of us, the ideaof zero may seem like a fairly easy and unchallenging thing to get our head around. But in actual fact, the ability to accept the absence of something, which is what zero is after all, as a quantity is a hard fought accomplishment. Children will often learn other numbers before understanding the concept of zero, and even then it has been shown that they can find difficulty in identifying whether or not zero is higher or lower than one.

So it is little wonder that in the animal world, very few species are known to understand the concept of zero. Chimpanzeesand some monkeyscan be trained to comprehend the concept, but apart from these, very few animals are able to do it, and until now it wasnt thought that any insect at all was capable of mastering the notion.

But scientists from the RMIT University in Melbourne have presented their research at the current Behaviour 2017 meeting in Portugal, reporting that they have been able to demonstrate that bees treat zero just like another number.

To start with, the researchers set up two platforms with varying numbers of shapes on them. They then trained bees to associate the platform with the fewest number of shapes on it with a sweet reward, and the platform with the most number a horrible taste. After being certain the bees were responding to the number of shapes and nothing else, the researchers then tested the insects by offering them one platform with two or three shapes, and another with zero shapes. The insects most frequently chose the latter.

Finally, the researchers then trained the bees to decide whether or not to land on a platform with zero objects, one object, or six objects. They found that once again, most of the time the insects could correctly identify the platform with nothing on it, but took more time over the decision if they were having to choose between a platform that had nothing and another that had just one object.

The fact that it took the bees longer to decide which one was zero when the numbers were numerically closer, suggests that the insects do indeed see the absence of objects as a number, the authors argue.

This would imply that the insects ability to count is similar to that of humans and some primates, and is strangely advanced for the animal world, not just for insects. The reason why bees should have such highly developed cognitive capability in the realm of mathematics, however, is a little tricky to deduce.

If you are getting stressed about upcoming exams then youre not alone, so is this artificially intelligent (AI) machine.

Last week, a top AI system was pitted against nearly 10 million students to face the maths paper for a much-feared Chinese university entrance exam, known as gaokao. Unfortunately for robotkind, its results were pretty mediocre.

The computer a humming tower of eleven servers with no Internet connection called AI-MATHS scored 105 points out of 150 points. On another version of the test, it scored 100. Although that beats the passing score of 90, humanities students had previously scored an average of 109 last year.

That said, the machine finished the exam in 10 minutes when humans are given two hours to complete the exam.

Scientists recently saidartificial intelligence will be able to beat humans at everything by 2060, whether that’squizzes, exams, chess, or the game Go. In response to the study, Elon Musk then tweeted that he believes AI-superiority will actually be earlier, around 2030 or 2040.

That doesnt mean this AI is slow off the mark, however. The computer itself would be able to deal with raw numbers with no problem. Instead, the purpose of this task was to understand the examination in terms of language, something that computers are not so sharp with at the moment.

“This is not a make-or-break test for a robot. The aim is to train artificial intelligence to learn the way humans reason and deal with numbers,” said Lin Hui, CEO of Chengdu Zhunxingyunxue Technology, who developed the AI, according to Chinese news agencyXinhua.

For example, the robot had a hard time understanding the words ‘students’ and ‘teachers’ on the test and failed to understand the question, so it scored zero for that question.

Gaokao isinfamously rigorous and renowned for being overwhelming stressful for the young people that take it. Made up of four three-hour papers in Chinese, English, mathematics, and a choice of either sciences or humanities, the series of tests rely on an extensive range of knowledge, problem-solving skills, and obscure creative thinking. The mathematics exam itself is said to be about as tough as the same level college exam in the West.

Nevertheless, the researchers continue to work with China’s Ministry of Science and Technology and remain optimistic their AI will improve in the exams in no time at all.

I hope next year the machine can improve its performance on logical reasoning and computer algorithms and score over 130,” Lin added.

When it comes to expressing a love for math, you can always count on Danica McKellar.

The 42-year-old actress, best known for playing Kevin Arnolds on-off girlfriend Winnie Cooper on The Wonder Years, has been keeping busy with her favorite subject.

My whole point, my whole mission is to make sure that kids never feel afraid of numbers. Never feel afraid of math, explained McKellar on FOX & Friends on Wednesday.

McKellar said plenty of people are intimidated by mathematics.

Theres this epidemic of kids being afraid of math, growing up that way, she said. Well, Im starting young. This is book one of eight that thatll be doing from Random House now, going through third grade. My mission is kids are never going to remember a time when numbers scare them.

She said her book is interactive.

But the fun thing about this book is that theres so many things to count on each page, she added. Ive snuck in a lot of math, including 10 frames on each page, to make your kids smarter.

McKellars new book, Goodnight, Numbers is the stars first picture book for children.

Statistics is a useful tool for understanding the patterns in the world around us. But our intuition often lets us down when it comes to interpreting those patterns. In this series we look at some of the common mistakes we make and how to avoid them when thinking about statistics, probability and risk.

You dont have to wait long to see a headline proclaiming that some food or behaviour is associated with either an increased or a decreased health risk, or often both. How can it be that seemingly rigorous scientific studies can produce opposite conclusions?

Nowadays, researchers can access a wealth of software packages that can readily analyse data and output the results of complex statistical tests. While these are powerful resources, they also open the door to people without a full statistical understanding to misunderstand some of the subtleties within a dataset and to draw wildly incorrect conclusions.

Here are a few common statistical fallacies and paradoxes and how they can lead to results that are counterintuitive and, in many cases, simply wrong.

Simpsons paradox

What is it?

This is where trends that appear within different groups disappear when data for those groups are combined. When this happens, the overall trend might even appear to be the opposite of the trends in each group.

One example of this paradox is where a treatment can be detrimental in all groups of patients, yet can appear beneficial overall once the groups are combined.

How does it happen?

This can happen when the sizes of the groups are uneven. A trial with careless (or unscrupulous) selection of the numbers of patients could conclude that a harmful treatment appears beneficial.

Example

Consider the following double blind trial of a proposed medical treatment. A group of 120 patients (split into subgroups of sizes 10, 20, 30 and 60) receive the treatment, and 120 patients (split into subgroups of corresponding sizes 60, 30, 20 and 10) receive no treatment.

The overall results make it look like the treatment was beneficial to patients, with a higher recovery rate for patients with the treatment than for those without it.

But note that the size and age distribution of each group is different between those who took the treatment and those who didnt. This is what distorts the numbers. In this case, the treatment group is disproportionately stacked with children, whose recovery rates are typically higher, with or without treatment.

Base rate fallacy

What is it?

This fallacy occurs when we disregard important information when making a judgement on how likely something is.

If, for example, we hear that someone loves music, we might think its more likely theyre a professional musician than an accountant. However, there are many more accountants than there are professional musicians. Here we have neglected that the base rate for the number of accountants is far higher than the number of musicians, so we were unduly swayed by the information that the person likes music.

How does it happen?

The base rate fallacy occurs when the base rate for one option is substantially higher than for another.

Example

Consider testing for a rare medical condition, such as one that affects only 4% (1 in 25) of a population.

Lets say there is a test for the condition, but its not perfect. If someone has the condition, the test will correctly identify them as being ill around 92% of the time. If someone doesnt have the condition, the test will correctly identify them as being healthy 75% of the time.

So if we test a group of people, and find that over a quarter of them are diagnosed as being ill, we might expect that most of these people really do have the condition. But wed be wrong.

In a typical sample of 300 patients, for every 11 people correctly identified as unwell, a further 72 are incorrectly identified as unwell.The Conversation, CC BY-ND

According to our numbers above, of the 4% of patients who are ill, almost 92% will be correctly diagnosed as ill (that is, about 3.67% of the overall population). But of the 96% of patients who are not ill, 25% will be incorrectly diagnosed as ill (thats 24% of the overall population).

What this means is that of the approximately 27.67% of the population who are diagnosed as ill, only around 3.67% actually are. So of the people who were diagnosed as ill, only around 13% (that is, 3.67%/27.67%) actually are unwell.

Worryingly, when a famous study asked general practitioners to perform a similar calculation to inform patients of the correct risks associated with mammogram results, just 15% of them did so correctly.

Will Rogers paradox

What is it?

This occurs when moving something from one group to another raises the average of both groups, even though no values actually increase.

The name comes from the American comedian Will Rogers, who joked that when the Okies left Oklahoma and moved to California, they raised the average intelligence in both states.

Former New Zealand Prime Minister Rob Muldoon provided a local variant on the joke in the 1980s, regarding migration from his nation into Australia.

How does it happen?

When a datapoint is reclassified from one group to another, if the point is below the average of the group it is leaving, but above the average of the one it is joining, both groups averages will increase.

Example

Consider the case of six patients whose life expectancies (in years) have been assessed as being 40, 50, 60, 70, 80 and 90.

The patients who have life expectancies of 40 and 50 have been diagnosed with a medical condition; the other four have not. This gives an average life expectancy within diagnosed patients of 45 years and within non-diagnosed patients of 75 years.

If an improved diagnostic tool is developed that detects the condition in the patient with the 60-year life expectancy, then the average within both groups rises by 5 years.

Berksons paradox can make it look like theres an association between two independent variables when there isnt one.

How does it happen?

This happens when we have a set with two independent variables, which means they should be entirely unrelated. But if we only look at a subset of the whole population, it can look like there is a negative trend between the two variables.

This can occur when the subset is not an unbiased sample of the whole population. It has been frequently cited in medical statistics. For example, if patients only present at a clinic with disease A, disease B or both, then even if the two diseases are independent, a negative association between them may be observed.

Example

Consider the case of a school that recruits students based on both academic and sporting ability. Assume that these two skills are totally independent of each other. That is, in the whole population, an excellent sportsperson is just as likely to be strong or weak academically as is someone whos poor at sport.

If the school admits only students who are excellent academically, excellent at sport or excellent at both, then within this group it would appear that sporting ability is negatively correlated with academic ability.

To illustrate, assume that every potential student is ranked on both academic and sporting ability from 1 to 10. There are an equal proportion of people in each band for each skill. Knowing a persons band in either skill does not tell you anything about their likely band in the other.

Assume now that the school only admits students who are at band 9 or 10 in at least one of the skills.

If we look at the whole population, the average academic rank of the weakest sportsperson and the best sportsperson are both equal (5.5).

However, within the set of admitted students, the average academic rank of the elite sportsperson is still that of the whole population (5.5), but the average academic rank of the weakest sportsperson is 9.5, wrongly implying a negative correlation between the two abilities.

This is where unexpected trends can occur through random chance alone in a data set with a large number of variables.

How does it happen?

When looking at many variables and mining for trends, it is easy to overlook how many possible trends you are testing. For example, with 1,000 variables, there are almost half a million (1,000×999/2) potential pairs of variables that might appear correlated by pure chance alone.

While each pair is extremely unlikely to look dependent, the chances are that from the half million pairs, quite a few will look dependent.

Example

The Birthday paradox is a classic example of the multiple comparisons fallacy.

In a group of 23 people (assuming each of their birthdays is an independently chosen day of the year with all days equally likely), it is more likely than not that at least two of the group have the same birthday.

People often disbelieve this, recalling that it is rare that they meet someone who shares their own birthday. If you just pick two people, the chance they share a birthday is, of course, low (roughly 1 in 365, which is less than 0.3%).

However, with 23 people there are 253 (23×22/2) pairs of people who might have a common birthday. So by looking across the whole group you are testing to see if any one of these 253 pairings, each of which independently has a 0.3% chance of coinciding, does indeed match. These many possibilities of a pair actually make it statistically very likely for coincidental matches to arise.

For a group of as few as 40 people, it is almost nine times as likely that there is a shared birthday than not.

The probability of no shared birthdays drops as the number of people in a group increases.The Conversation, CC BY-ND

Just like everyone knows you should never wake sleepwalkers, bulls hate red and Napoleon was short.

Wrong on all counts. Waking sleepwalkers will cause them no harm in fact, theyre more likely to harm themselves while sleepwalking. Bulls are colorblind; theyre attracted to movement. And Napoleon was 57, which was above average height for Frenchman during his lifetime.

So why do we believe that American public schools are doing such a terrible job?

We have made a commitment to every single child regardless of what their parents can afford to pay, regardless of their access to transportation, regardless of whether they can afford uniforms, lunch or even if they have a home. Heck! We even provide education to children who are here illegally.

We define education differently. Though our laws are woefully backward, in practice we look at it as a right, not a privilege. And for a full 13 years (counting kindergarten) its a right for every child, not just some.

But thats not all! We also provide some of the highest quality education you can get in the world! We teach more, help more, achieve more and yet we are criticized more than any system in any country in the world.

TEST SCORES

Critics argue that our scores on international tests dont justify such a claim. But theyre wrong before you even look at the numbers. Theyre comparing apples to pears. You simply cant compare the United States to countries that leave hundreds of thousands of rural and poor children without any education whatsoever. The Bates Motel may have the softest pillows in town, but its immediately disqualified because of the high chance of being murdered in the shower.

No school system of this size anywhere in the world exceeds the United States in providing free access to education for everyone. And that, alone, makes us one of the best.

It doesnt mean our system is problem free. There are plenty of ways we could improve. Were still incredibly segregated by race and class. Our funding formulas are often regressive and inadequate. Schools serving mostly poor students dont have nearly the resources of those serving rich students. But at least at the very outset what were trying to do is better than what most of the world takes on. You cant achieve equity if it isnt even on the menu.

However, for some people, this will not be enough. Theyll say that despite our high ideals, the quality of what we actually provide our students is low. After all, those international test scores are so low.

First point: it depends on the scores youre looking at. American elementary and middle school students have improved on theTrends in International Mathematics and Science Study every four years since the tests began in 1995. They are above the international average in all categories and within a few percentage points of the global leaders (something rarely mentioned on the nightly news).

Thats why many of these countries where the poorest children do not have access to education have higher test scores than the United States. Youre not comparing equals. The United States has the highest child poverty rate in the Western World. And we dont hide them away. We include them on our tests. That has a major impact on our scores. But talking heads on TV almost always ignore it. They pretend it doesnt exist. Its the only way they can use these test scores to prove to a gullible audience that Americas schools are failing.

But if you fairly compare education systems and factor in the equal access we provide for all children to an education, our system comes out way on top. We have one of the best systems in the world.

But wait! Theres more!

SPECIAL EDUCATION

Not only does the United States serve all children regardless of academic achievement or poverty. We also serve far more students with disabilities.

Why are there so many special education children in the USA? Because we have a higher standard of living.

A standard pregnancy lasts about 280 days or 40 weeks. However, some mothers give birth to children after only 28 weeks. Two decades ago, these babies would not have survived. Today, they often do. Five years later that child will enter kindergarten and our school system will be responsible for teaching that student to read, write and learn math. In other countries, premature babies have a much lower chance of survival. They dont survive to become the special education population. So things as diverse as the live-birth rate actually affect average test scores.

Would you say this makes other countries superior to the United States? Heck no! In fact, just the opposite. I certainly wouldnt wish more underperforming U.S. students were ending their lives so we could do better on international tests. Nor would I wish that more premature babies died to improve our international standing.

We have developed a special education system to help children at the edges that many other countries just cant touch. In some countries these students are simply excluded. In others they are institutionalized. In some countries its up to parents to find ways to pay for special services. The United States is one of the only countries where these children are not only included and offered full and free access, but the schools go above and beyond to teach these children well beyond their 12th academic year.

In every public school in the United States these students are included. In math, reading, science and social studies, they are there benefiting from instruction with the rest of the class. And this, in turn, benefits even our non-special education students who gain lessons in empathy and experience the full range of human abilities.

Of course, most of our special education students are also included in our test scores. Yes, other countries that ignore these children and exclude them from testing get higher scores. But so what? Do you mean to tell me this makes them better? No, it makes them worse.

In many ways, we are the gold standard, not them. They should be emulating us, not the other way around. They should be jealous of the way we prize each others humanity. We shouldnt be salivating at test scores achieved through shunning certain students in favor of others.

The bottom line: the curriculum at most American schools is more inclusive than that found internationally. We even include societal issues like alcohol and drug abuse prevention, stress reduction and relaxation, and physical fitness programs.

In addition we dont stratify our children based on academic ability to nearly the same degree as many international schools. We dont weed out our worst students through middle and high school until only our most capable are left in 12th grade. Nor is college only open to our best and brightest. We make a much greater effort than many other countries to keep this option open to as many students as possible regardless of whether they can afford it or not. The number of Americans with at least some college educationhas soared over the past 70 years, from 10 percent in 1940 to 56 percent today, even as the population has tripled and the nation has grown vastly more diverse. Meanwhile, Graduation rates are at an all-time high of 83.2 percent, and for the first time minority students are catching up with their white counterparts.

Its not easy. But its something were committed to as a nation. And thats not true around the world.

SIZE MATTERS

Its much easier to educate less children. Even excellent education systems would struggle with our sheer numbers. Small systems often outshine bigger ones. For instance, I might be able to make dinner for my immediate family, but Id find it much more challenging to prepare a meal for a banquet hall of hundreds. Similarly, it remains to be seen whether smaller nations could handle educating a population as big and diverse as ours without collapsing.

By any fair measure, Americas public education system is simply stunning. But the media perpetuates the myth that were failing.

PUBLIC PERCEPTION AND THE MEDIA

After decades of hearing these falsehoods, the American public is strikingly divided. On a 2011 Gallup poll, parents were asked their opinion of their local school and the public was asked its opinion of schools in general. The results are enlightening. Parents who gave their local school an A grade were at the highest percentage ever (37%) whereas only 1% of respondents rated the nations schools that way. Why the difference? Respondents said it was mostly because people knew about their local schools through direct experience. They only learned about the state of education nationally through the news media.

But even when journalists want to be fair, its difficult for them to get the inside story of how our public schools work. They are rarely permitted inside our schools to see the day-to-day classroom experience. Legal issues about which students may be photographed, filmed or interviewed, the difficulty of getting parental permissions and the possibility of embarrassment to principals and administrators often keeps the doors closed. In many districts, teachers arent even allowed to speak on the record to the media or doing so can make them a political target. So reporters are often in the position of being unable to directly experience the very thing theyre reporting on. Imagine if sportswriters never got to see athletes play or political reporters never attended a campaign rally. Of course there would be a disconnect!

So were left with a public education system that should be the envy of the world being portrayed as a loser.

Lets assume that Bolt won the race by 10m from the second placed runner, Justin Gatlin. And now lets run the final again, but this time Bolt will start 10m behind the start line. If both Bolt and Gatlin run at the same constant speed in the second race as they did in the first, who wins?

2) The mystery games

(For the purposes of this puzzle, any resemblance to real persons, living or dead, and Olympic sports, actual or obsolete, is purely coincidental.)

Nafissatou, Jessica, and Brianne are taking part in an athletics competition that involves at least two events. In each event the winner gets G points, second placed gets S points and third place gets B points, where G, S and B are whole numbers, nonzero, and G> S>B. No event is tied. Nafissatou scores 22 points in total. Jessica and Brianne score 9 points each in total. Jessica wins the 100m hurdles. Who is second in the javelin? How many events are there?

Ill open comments at noon, and post the answers at 5pm BST. Please dont post the answer in the comments before 5pm because this spoils it for the many people who want to work out the answers themselves. Thanks!

I post a puzzle here on a Monday every two weeks. If you want to propose a puzzle for this column, please email me Id love to hear it.

Youll probably remember from your math lessons at school that you cant divide a number by zero, even if you didnt understand why exactly. If you try to divide a number by zero on a electronicpocket calculator, itll pop upwith a message saying Error. If you try this on a mechanical calculator, it looks like youve ripped an irreversible hole in the fabric ofspace-time.

This video by MultiGlizda shows the chaos that unfolds when you try to divide by zero on a Facit ESA-01 mechanical calculator with its casing off. Not only does it give an interesting insight into how these old calculators work, it also reveals the slippery nature of the number zero and its division.

As YouTube channel Numberphile explains, division is based on subtraction. If you want to divide a number by a second number, you simply subtract the second number from the first over and over again. For example, 20 divided by 5 would be: 20 – 5 = 15, 15 – 5 = 10, 10 – 5 = 5, 5 – 5 = 0. Since that took 4 subtractions, the answer is 4.

However, divide 20 by 0 and youll end up subtracting 0 from 20 an infinite amount of times. While that doesnt mean that 0 equals infinity, it appears that the mechanical calculator is attempting to complete the infinite number of operations it believes it needs to complete the division. You can check out this Numberphile video if you want to know more about dividing by zero and why it doesnt equal infinity.

Meanwhile, some say the Facit ESA-01 is still flicking through numbers to this day.

The gender gap in STEM (science, technology, engineering and mathematics) is widely reported. Only one-quarter of college graduates entering careers in STEM in the U.S. are women. The numbers are even more dismal in fields like physics and engineering. Only about 1 in 10 physicists and astronomers are women. About 8 percent of mechanical engineers are female.

But here’s the rub: Girls are just as interested and are definitely not less skilled in STEM subjects than boys. In fourth grade, both genders report similar rates of interest in science. From K-12, female and male students generally perform equally well on standardized math and science tests. High school boys and girls also enroll in advanced science courses at comparable rates.

Ellis et al

So, what exactly happens to all these STEM-loving girls?

There are many “leaks” in the so-called STEM pipeline, including educational shortfalls and cultural issueslike stereotyping. But there may be one issue in particular that’s having a profound impact on the number of women in STEM: the notoriously difficult college math class, Calculus I, a new study from Colorado State University finds.

Male and female students lose confidence in their math skills at a similar rate in Calc I. But women are far more likely to be discouraged by the class, a necessary process for those pursuing a career in STEM, according to the report, published in the journal PLOS ONE last month.

Researchers found that female students have the same level of academic preparedness and similar career goals as their male counterparts, yet they’re 1.5 times more likely than men to leave their STEM studies after taking Calc I.

Part of the problem is that women enter the class with less confidence in the first place. When comparing men and women with above-average mathematical abilities, the researchers found that female students had significantly lower mathematical confidence both at the start and the end of the college term.

The findings suggest that a major factor in women’s decisions to leave STEM paths after Calculus I have nothing to do with ability, but confidence in their ability. (Though this particular study did not examine students’ grades in the class, a 2015 paper about college math concluded that women outperform men in Calc I.)

“When women are leaving, it is because they don’t think they can do it – not because they can’t do it,” said study co-author Bailey Fosdick in a press release.

Fixing this leak at the “Calculus I juncture” of the STEM pipeline could have an extraordinary effect on the number of women continuing their studies and getting jobs in the field, the researchers said.

“Our findings indicate that if women persisted in STEM at the same rate as men starting in Calculus I, the number of women entering the STEM workforce would increase by 75 percent,” the study reads.

This boost could go a long way, researchers said, in fulfilling the need for more STEM workers in the U.S. The Obama administration has said that 1 million additional STEM graduates will be needed to fulfill demand by 2022.

To plug the leak, the researchers said a multi-pronged approach will be necessary to raise the confidence in all STEM students. Improving teaching quality and encouraging students will be critical first steps.

“If female students are entering college excited to be challenged, supported, and surrounded by like-minded STEM people and they have a negative initial experience with a STEM course, it makes sense that this could be the final experience to encourage them to pursue a different (and non-STEM) field,” Jessica Ellis, a study co-author and professor of mathematics at Colorado State University, told Vocativ on Tuesday.

Ellis said she tries to “raise the confidence” of all students in her classes. Since the study, she said she’s made a special effort “to make sure women have a voice and if they get something wrong once, to let them know that’s good and not bad.”